Optimal Rate for Sequential Marginal Calibration

Online Forecasting vs Adversarial Outcomes | Collina et al., arXiv:2601.05245

T^{0.54}
Known Lower Bound
T^{2/3}
Known Upper Bound
0.983
Best Empirical Exp.
10K
Max Horizon

Calibration Error Scaling (Log-Log)

Adversary Comparison (T=2000)

Scaling Exponents

Summary

ForecasterExponentR^2Strategy
Fixed Grid0.9951.000Static grid
Adaptive Grid0.9831.000Refining grid
Randomized Rounding1.0001.000Probabilistic
Minimax Potential1.0001.000Potential-based